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A question to practice functions, graphs and domains

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Functons, finding the domain and range of functons

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Click 'Try another question like this one' if you need more practice.

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Find the domain of function \\begin{align} y &= \\frac{\\simplify {{m2}x+{c2}}}{ \\simplify {{m}x+{c}} }\\text{.}
\\end{align}

Domain: all $x$ except [[0]]

Find the domain and range of function \\begin{align} y &= \\simplify{{-m5}} - \\sqrt{\\simplify {{m3}x+{c3}}}\\text{.}
\\end{align}

Put \">=\" for \"$\\ge$\" and \"<=\" for \"$\\le$\", otherwise \">\" or \"<\".

Domain: all $x$ satisfying $x$ [[2]] [[0]]

Range: all $y$ satisfying $y$ [[3]] [[1]]

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Find the domain of function \\begin{align}y &= \\sqrt{\\frac{\\simplify {{c4}}}{ \\simplify {{m22}x+{c2}} }}\\text{.}
\\end{align}

Domain: all $x$ satisfying $x$[[1]][[0]]

Multiplication and adding matrices.

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To add matrices, just add corresponding elements together.

a)

\\begin{align}
\\mathbf{AB} &= \\var{A}\\var{B} \\\\
&= \\begin{pmatrix} \\simplify[]{ {a[0][0]}*{b[0][0]}+{a[0][1]}*{b[1][0]} } & \\simplify[]{ {a[0][0]}*{b[0][1]} + {a[0][1]}*{b[1][1]} } \\\\ \\simplify[]{ {a[1][0]}*{b[0][0]} + {a[1][1]}*{b[1][0]} } & \\simplify[]{ {a[1][0]}*{b[0][1]} + {a[1][1]}*{b[1][1]} } \\end{pmatrix} \\\\
&= \\var{a*b}
\\end{align}

b)

\\begin{align}
\\mathbf{BA} &= \\var{B}\\var{A} \\\\
&= \\begin{pmatrix} \\simplify[]{ {b[0][0]}*{a[0][0]}+{b[0][1]}*{a[1][0]} } & \\simplify[]{ {b[0][0]}*{a[0][1]} + {b[0][1]}*{a[1][1]} } \\\\ \\simplify[]{ {b[1][0]}*{a[0][0]} + {b[1][1]}*{a[1][0]} } & \\simplify[]{ {b[1][0]}*{a[0][1]} + {b[1][1]}*{a[1][1]} } \\end{pmatrix} \\\\
&= \\var{b*a}
\\end{align}

c)

\\begin{align}
\\mathbf{CB} &= \\var{C}\\var{B} \\\\
&= \\begin{pmatrix} \\simplify[]{ {c[0][0]}*{b[0][0]}+{c[0][1]}*{b[1][0]} } & \\simplify[]{ {c[0][0]}*{b[0][1]} + {c[0][1]}*{b[1][1]} } \\\\ \\simplify[]{ {c[1][0]}*{b[0][0]} + {c[1][1]}*{b[1][0]} } & \\simplify[]{ {c[1][0]}*{b[0][1]} + {c[1][1]}*{b[1][1]} } \\end{pmatrix} \\\\
&= \\var{c*b}
\\end{align}

d)

\\begin{align}
\\mathbf{AC} &= \\var{A}\\var{C} \\\\
&= \\begin{pmatrix} \\simplify[]{ {a[0][0]}*{c[0][0]}+{a[0][1]}*{c[1][0]} } & \\simplify[]{ {a[0][0]}*{c[0][1]} + {a[0][1]}*{c[1][1]} } \\\\ \\simplify[]{ {a[1][0]}*{c[0][0]} + {a[1][1]}*{c[1][0]} } & \\simplify[]{ {a[1][0]}*{c[0][1]} + {a[1][1]}*{c[1][1]} } \\end{pmatrix} \\\\
&= \\var{a*c}
\\end{align}

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Given matrices $A = \\var{A1}, B=\\var{B1}, C=\\var{C1}$:

1) Identify pairs that can be added:[[0]]

2) Identify pairs that cannot be multiplied:[[1]]

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Given matrices $\\mathbf{A}$ and $\\mathbf{B}$ choose the correct matrix size and calculate:

1) $\\mathbf{A+B} = \\var{A1}+\\var{C1} = $ [[0]]

2) $\\mathbf{A \\times B} = \\var{A1}\\var{B1} = $ [[1]]

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